hierarchy, intended as slight weakenings of the [[Kunen inconsistency]], but similar in spirit.

hierarchy, intended as slight weakenings of the [[Kunen inconsistency]], but similar in spirit.

−

The ''wholeness axioms'' are formalized in the

+

The ''wholeness axioms'' are formalized in the language $\{\in,j\}$, augmenting the usual language of set theory $\{\in\}$ with an additional unary function symbol $j$ to represent the [[elementary embedding]]. The base theory ZFC is expressed only in the smaller language $\{\in\}$. Corazza's

−

language $\{\in,j\}$, augmenting the usual language of set

+

original proposal, which we denote by $\text{WA}_0$, asserts that $j$ is a nontrivial amenable elementary embedding from the universe to itself, without adding formulas containing $j$ to the separation and replacement axioms. Elementarity is expressed by the scheme $\varphi(x)\iff\varphi(j(x))$, where $\varphi$ runs through the formulas of the usual language of set theory; nontriviality is expressed by the sentence $\exists x j(x)\not=x$; and amenability is simply the assertion that $j\upharpoonright A$ is a set for every set $A$. Amenability in this case is equivalent to the assertion that the separation axiom holds for $\Delta_0$ formulae in the language $\{\in,j\}$. The wholeness axiom $\text{WA}$, also denoted $\text{WA}_\infty$, asserts in addition that the full separation axiom holds in the language $\{\in,j\}$.

−

theory $\{\in\}$ with an additional unary function symbol $j$

+

−

to represent the [[elementary embedding]]. The base theory ZFC is

+

−

expressed only in the smaller language $\{\in\}$. Corazza's

+

−

original proposal, which we denote by $\text{WA}_0$, asserts

+

−

that $j$ is a nontrivial amenable elementary embedding

+

−

from the universe to itself. Elementarity is expressed by

+

−

the scheme $\varphi(x)\iff\varphi(j(x))$, where $\varphi$

+

−

runs through the formulas of the usual language of set

+

−

theory; nontriviality is expressed by the sentence $\exists

+

−

x j(x)\not=x$; and amenability is simply the assertion

+

−

that $j\upharpoonright A$ is a set for every set $A$. Amenability in this case is equivalent to

+

−

the assertion that the separation axiom holds for

+

−

$\Delta_0$ formulae in the language $\{\in,j\}$.

+

−

The wholeness axiom $\text{WA}$, also denoted $\text{WA}_\infty$, asserts in addition that the

+

−

full separation axiom holds in the language $\{\in,j\}$.

+

−

Those two axioms are the endpoints of the hierarchy of axioms $\text{WA}_n$, asserting increasing amounts of the separation axiom.

+

Those two axioms are the endpoints of the hierarchy of axioms $\text{WA}_n$, asserting increasing amounts of the separation axiom. Specifically, the wholeness axiom $\text{WA}_n$, where $n$ is amongst $0,1,\ldots,\infty$, consists of the following:

−

Specifically, the wholeness axiom $\text{WA}_n$, where $n$ is

+

−

amongst $0,1,\ldots,\infty$, consists of the following:

+

# (elementarity) All instances of $\varphi(x)\iff\varphi(j(x))$ for $\varphi$ in the language $\{\in,j\}$.

# (elementarity) All instances of $\varphi(x)\iff\varphi(j(x))$ for $\varphi$ in the language $\{\in,j\}$.

Line 30:

Line 13:

# (nontriviality) The axiom $\exists x\,j(x)\not=x$.

# (nontriviality) The axiom $\exists x\,j(x)\not=x$.

−

Clearly, this resembles the [[Kunen inconsistency]]. What is missing from the wholeness

+

Clearly, this resembles the [[Kunen inconsistency]]. What is missing from the wholeness axiom schemes, and what figures prominantly in Kunen's

−

axiom schemes, and what figures prominantly in Kunen's

+

proof, are the instances of the replacement axiom in the full language with $j$. In particular, it is the replacement axiom in the language with $j$ that allows one to define the critical sequence $\langle \kappa_n\mid n\lt\omega\rangle$, where $\kappa_{n+1}=j(\kappa_n)$, which figures in all the proofs of the Kunen inconsistency. Thus, none of the proofs of the Kunen inconsistency can be carried out with $\text{WA}$, and indeed, in every model of $\text{WA}$ the critical sequence is unbounded in the ordinals.

−

proof, are the instances of the replacement axiom in the

+

−

full language with $j$. In particular, it is the replacement axiom in the language with $j$ that allows one to define the critical sequence $\langle \kappa_n\mid n\lt\omega\rangle$, where $\kappa_{n+1}=j(\kappa_n)$, which figures in all the proofs of the Kunen inconsistency. Thus, none of the proofs of the Kunen inconsistency can be carried out with WA, and indeed, in every model of $\text{WA}$ the critical sequence is unbounded in the ordinals.

+

The hiearchy of wholeness axioms is strictly increasing in strength, if consistent. <cite>Hamkins2001:WholenessAxiomAndVequalHOD</cite>

The hiearchy of wholeness axioms is strictly increasing in strength, if consistent. <cite>Hamkins2001:WholenessAxiomAndVequalHOD</cite>

Revision as of 13:28, 8 December 2017

The wholeness axioms, proposed by Paul Corazza [1, 2], occupy a
high place in the upper stratosphere of the large cardinal
hierarchy, intended as slight weakenings of the Kunen inconsistency, but similar in spirit.

The wholeness axioms are formalized in the language $\{\in,j\}$, augmenting the usual language of set theory $\{\in\}$ with an additional unary function symbol $j$ to represent the elementary embedding. The base theory ZFC is expressed only in the smaller language $\{\in\}$. Corazza's
original proposal, which we denote by $\text{WA}_0$, asserts that $j$ is a nontrivial amenable elementary embedding from the universe to itself, without adding formulas containing $j$ to the separation and replacement axioms. Elementarity is expressed by the scheme $\varphi(x)\iff\varphi(j(x))$, where $\varphi$ runs through the formulas of the usual language of set theory; nontriviality is expressed by the sentence $\exists x j(x)\not=x$; and amenability is simply the assertion that $j\upharpoonright A$ is a set for every set $A$. Amenability in this case is equivalent to the assertion that the separation axiom holds for $\Delta_0$ formulae in the language $\{\in,j\}$. The wholeness axiom $\text{WA}$, also denoted $\text{WA}_\infty$, asserts in addition that the full separation axiom holds in the language $\{\in,j\}$.

Those two axioms are the endpoints of the hierarchy of axioms $\text{WA}_n$, asserting increasing amounts of the separation axiom. Specifically, the wholeness axiom $\text{WA}_n$, where $n$ is amongst $0,1,\ldots,\infty$, consists of the following:

(elementarity) All instances of $\varphi(x)\iff\varphi(j(x))$ for $\varphi$ in the language $\{\in,j\}$.

(separation) All instances of the Separation Axiom for $\Sigma_n$ formulae in the full language $\{\in,j\}$.

(nontriviality) The axiom $\exists x\,j(x)\not=x$.

Clearly, this resembles the Kunen inconsistency. What is missing from the wholeness axiom schemes, and what figures prominantly in Kunen's
proof, are the instances of the replacement axiom in the full language with $j$. In particular, it is the replacement axiom in the language with $j$ that allows one to define the critical sequence $\langle \kappa_n\mid n\lt\omega\rangle$, where $\kappa_{n+1}=j(\kappa_n)$, which figures in all the proofs of the Kunen inconsistency. Thus, none of the proofs of the Kunen inconsistency can be carried out with $\text{WA}$, and indeed, in every model of $\text{WA}$ the critical sequence is unbounded in the ordinals.

The hiearchy of wholeness axioms is strictly increasing in strength, if consistent. [3]

If $j:V_\lambda\to V_\lambda$ witnesses a rank into rank cardinal, then $\langle V_\lambda,\in,j\rangle$ is a model of the wholeness axiom.

If the wholeness axiom is consistent with $\text{ZFC}$, then it is consistent with $\text{ZFC+V=HOD}$.[3]